Calculus Examples

$\frac{lim_{x \to 0} \frac{1}{x + 4} - \frac{1}{4}}{x}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \frac{1}{x + 4} - lim_{x \to 0} \frac{1}{4}}{x}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{lim_{x \to 0} 1}{lim_{x \to 0} x + 4} - lim_{x \to 0} \frac{1}{4}}{x}$

Step 1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{lim_{x \to 0} x + 4} - lim_{x \to 0} \frac{1}{4}}{x}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\frac{1}{lim_{x \to 0} x + lim_{x \to 0} 4} - lim_{x \to 0} \frac{1}{4}}{x}$

Step 1.5

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{lim_{x \to 0} x + 4} - lim_{x \to 0} \frac{1}{4}}{x}$

Step 1.6

Evaluate the limit of $\frac{1}{4}$ which is constant as $x$ approaches $0$ .

$\frac{\frac{1}{lim_{x \to 0} x + 4} - \frac{1}{4}}{x}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\frac{1}{0 + 4} - \frac{1}{4}}{x}$

Step 3

Simplify the answer.

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Step 3.1

Add $0$ and $4$ .

$\frac{\frac{1}{4} - \frac{1}{4}}{x}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Combine the numerators over the common denominator.

$\frac{\frac{1 - 1}{4}}{x}$

Step 3.2.2

Subtract $1$ from $1$ .

$\frac{\frac{0}{4}}{x}$

Step 3.2.3

Divide $0$ by $4$ .

$\frac{0}{x}$

Step 3.3

Divide $0$ by $x$ .

$0$